A compactness result in $GSBV^p$ and applications to $\Gamma$-convergence for free discontinuity problems

TL;DR

This paper extends compactness results in the space GSBV^p to problems without initial bounds, and applies these results to establish Gamma-convergence for free discontinuity problems, including convergence of minima and minimizers.

Contribution

It introduces a new compactness theorem in GSBV^p without a priori bounds and applies it to Gamma-convergence analysis of free discontinuity problems.

Findings

Extended compactness in GSBV^p without bounds

Proved Gamma-convergence for boundary value problems

Showed convergence of minimum values and minimizers

Abstract

We present a compactness result in the space $GSBV^p$ which extends the classical statement due to Ambrosio to problems without a priori bounds on the deformations. As an application, we revisit the $\Gamma$-convergence results for free discontinuity functionals established recently by Cagnetti, Dal Maso, Scardia, and Zeppieri. We investigate sequences of boundary value problems and show convergence of minimum values and minimizers.